Diffusive excitonic bands from frustrated triangular sublattice in a singlet-ground-state system

Magnetic order in most materials occurs when magnetic ions with finite moments arrange in a particular pattern below the ordering temperature. Intriguingly, if the crystal electric field (CEF) effect results in a spin-singlet ground state, a magnetic order can still occur due to the exchange interactions between neighboring ions admixing the excited CEF levels. The magnetic excitations in such a state are spin excitons generally dispersionless in reciprocal space. Here we use neutron scattering to study stoichiometric Ni2Mo3O8, where Ni2+ ions form a bipartite honeycomb lattice comprised of two triangular lattices, with ions subject to the tetrahedral and octahedral crystalline environment, respectively. We find that in both types of ions, the CEF excitations have nonmagnetic singlet ground states, yet the material has magnetic order. Furthermore, CEF spin excitons from the tetrahedral sites form a dispersive diffusive pattern around the Brillouin zone boundary, likely due to spin entanglement and geometric frustrations.

NiZnMo3O8 will be compared with Ni2Mo3O8 to identify the CEF ground states of both materials in discussions to follow. The neutron powder diffraction experiments on NiZnMo3O8 were performed at room temperature using the high-resolution powder diffractometer BT-1, at NCNR. 5 grams of powder were used. The refined structure has a space group P63mc with lattice parameter a = b = 5.761 Å, c = 9.830 Å. The positions and occupation fractions are refined, yielding 11% disorder for tetrahedral and octahedra sites. The fitting results an R1 = 4.78%. Supplementary Figure 2 shows the refinement results from powder neutron diffraction experiments. Supplementary Table 1 shows the refinement results to be compared with Table 1 for Ni2Mo3O8 shown in the main text.
One piece of single crystalline Ni2Mo3O8 was aligned in the [H, 0, L] zone at CORELLI, BL-9, SNS. Magnetic Bragg peaks which can be indexed as (1/2,0,0) are clear in the diffraction pattern (inset of Fig. 1e). The magnetic structure of Ni2Mo3O8 was determined from the refinement of 32 magnetic Bragg peaks using the FullProf program. This is consistent with the previous report, except the exact magnetic moments perpendicular and parallel to the c-axis are different. The outcome of our refinement is shown in Supplementary

III. Susceptibility Measurements with Magnetic Fields along the c-axis
Supplementary Figure 3. The temperature dependence of magnetic susceptibility (a) and inverse susceptibility (b) measured with magnetic fields applied along the c-axis.

IV. Crystal Electrical Field (CEF) Analysis
We consider the simplest undistorted cubic crystal field (CF) and the spin orbit coupling (SOC); the level scheme would be: Octahedral site: Tetrahedral CF: 3T1 (S=1, L=1 at twice the energy of the 3T2 level but with zero dipole intensity) 3T2 (S=1, L=1 orbital triplet excited state at around 200 meV (above 100-500 meV) 3A2 (S=1, L=0 orbital singlet ground state) SOC does not split the 3A2 state. To identify the scattering around 17 meV, we carried out high-energy CEF levels measurements on Ni2Mo3O8, NiZnMo3O8, and Zn2Mo3O8 at 4.5 K using incident neutron energy Ei = 40 meV and 250 meV on SEQUOIA, BL-17, SNS (Supplementary Figure 4). The low-energy CEF measurements on Ni2Mo3O8 and NiZnMo3O8 were performed at 2.0 K with Ei = 3.7 meV at LET, ISIS (Supplementary Figure 5). We used polycrystalline samples of Ni2Mo3O8 (6.0 grams), NiZnMo3O8 (4.24 grams), and Zn2Mo3O8 (4.33 grams).
Supplementary Figure 4. Powder spectra and constant-wavevector cut of Ni2Mo3O8, NiZnMo3O8, and Zn2Mo3O8 on SEQUOIA, SNS. a-c, Spin excitation spectra of Ni2Mo3O8, NiZnMo3O8, and Zn2Mo3O8 at 4.5 K with Ei = 40 meV. d-f, Spin excitation spectra of Ni2Mo3O8, NiZnMo3O8, and Zn2Mo3O8 at 4.5 K with Ei = 250 meV. g, Constant-wavevector cuts of panel ac, which show that the intensity is dramatically reduced when the tetrahedral Ni site is occupied by non-magnetic Zn.
Supplementary Figure 5. Low-energy powder spectra of Ni2Mo3O8 and NiZnMo3O8 on LET, ISIS, and heat capacity data of NiZnMo3O8. a-b, Spin excitation spectra of Ni2Mo3O8 and NiZnMo3O8 at 2 K with Ei = 3.7 meV. c, Specific heat of NiZnMo3O8 measured on a single crystal with 0-7 T magnetic fields perpendicular to the c-axis.
Since the intensity of the scattering around 17 meV drops to about 1/10 when about 90% tetrahedral site Ni is replaced by non-magnetic Zn in NiZnMo3O8 [ Supplementary Figure 4g], we conclude that the flat band scattering at 17 meV is totally from the tetrahedral coordinated Ni sites [Supplementary Figure 6a]. Besides, we found at least 3 peaks in the cut (Fig. 2c in the main text) indicating that the es here is from a doublet. Therefore, both tetrahedral and octahedral Ni sites have a singlet gs and doublet es, consistent with previous predictions [1].
Qualitatively, the singlet-doublet splitting modes for the octahedral site are in low energy region, as the low energy scattering in both Ni2Mo3O8 and NiZnMo3O8 are below 1.5 meV. We fit the heat capacity of NiZnMo3O8 to a two-level Schottky anomaly model: Based on the point-group symmetry at the Ni 2+ atomic site and using the Stevens operator formalism, the Hamiltonian of the CEF for both sites is = 2 0 Ô 2 0 + 4 0̂4 0 + 4 3̂4 3 , where (m and n are integers and ≥ ) are CEF parameters that will be determined experimentally, and the Stevens operators ̂n m are polynomial functions of the components of the total angular momentum operator J z , J + , and J − (J ± = ± ).
For the octahedral site, we fit the CEF parameters to the 0.8 meV spin gap and magnetic susceptibility data of NiZnMo3O8, resulting 2 0 = 1.0 , 4 0 = − 0.5 , and 4 3 = 30 , which gives the first excited doublet at 0.78 meV and the second excited doublet at 634 meV. The fitting for the tetrahedral site is more sophisticated due to the intervention of two magnetic Ni sites in the susceptibility data of Ni2Mo3O8. A rough fit that mainly depends on the 17 meV spin gap yields 2 0 = 7.0 , 4 0 = − 0.7 , and 4 3 = 2.2 , with the first excited doublet at 17.0 meV and the second excited doublet at 201 meV.

L-dependence and Temperature dependence of spin excitations in the high energy range Supplementary Figure 7. Momentum and temperature dependence of magnetic scattering in
Ni2Mo3O8 measured with the assembly of single crystals on SEQUOIA. a-c, Momentum dependence of the magnetic scattering at 1.5 K, 10 K, and 120 K corresponding to Fig. 3i, 3j, and 3k, respectively. The energy integration range is 12 meV -16 meV. i-k, Momentum dependence of the magnetic scattering at 1.5 K, 10 K, and 120 K corresponding to Fig. 3f, 3g, and 3h, respectively. The energy integration range is 18 meV -22 meV. Data in both energy ranges show no L-modulation, indicating a 2D effective mode for high-energy spin excitations.
Supplementary Figure 7 shows the L-dependence of spin excitations for the tetrahedral sites, revealing that spin excitations are weakly L-dependent and better defined above the TN at 10 K. This is consistent with the notion that spin excitations in these energy ranges are 2D.  38, 7.70, 8.46, and 11.01 Å for panels a-d.

VII. Cluster Form Factor Calculation
Supplementary Figure 10. Schematic of a spin cluster and * 2 for 120° and FM configurations.
We will discuss 120° and ferromagnetic (FM) configurations in this section. We first consider a 120° spin configuration on 3 Ni atoms sitting on the orthorhexagonal lattice with the following coordinates: at ( ( The Fourier transform of the spin on this cluster is: